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[[52,2,7]] d ≤
n
52
k
2
d
7
kd²/n
1.885
w
4
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 7, d_Z ≤ 7 · w_X = 4, w_Z = 4 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 7 · witness weight 7 (claimed upper_bound)
witness operator (support, 7 qubits)
[0, 9, 12, 16, 25, 32, 41]
d_Z 7 · witness weight 7 (claimed upper_bound)
witness operator (support, 7 qubits)
[5, 14, 30, 33, 37, 39, 47]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

Diagnostics

computed by the verifier from the parity checks, the layout, and the stored witnesses; shown as evidence, not used for ranking
girth H_X 8 · H_Z 8 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 4 · H_Z 4
qubit degrees H_X 1–2 (mean 1.923) · H_Z 1–2 (mean 1.923)
trapping sets H_X (1,1)×4 (2,1)×12 (3,1)×36 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 4 (1,2): 48 (2,1): 12 (2,2): 138 (3,1): 36 (3,2): 396 (3,3): 12 (3,4): 88
trapping sets H_Z (1,1)×4 (2,1)×12 (3,1)×36 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 4 (1,2): 48 (2,1): 12 (2,2): 138 (3,1): 36 (3,2): 396 (3,3): 12 (3,4): 88

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction Reconstructed from qecdb.org record 67bc8f4b65e5efe366a705cb (https://qecdb.org/codes/67bc8f4b65e5efe366a705cb); the record's H field is read as binary symplectic rows and split by rowspace (rank(A|B) = rank(A) + rank(B)), keeping the lightest independent basis of each part; n, k and the check weight recomputed here, witnesses from the kit's random search at the trials below. Database description: 2BGA; database distance claim d=7 (lower 7, upper 7).
model Mimo-V2.6-Free (claimed, not verified)
date 2026-09-27
notes Equivalence assessed against the board: no exact stabilizer-group fingerprint match and no Weisfeiler-Leman signature match to any existing entry.
family generalized bicycle (a tag, not a ranking)
locality unrestricted (computed from the layout)
weight class weight ≤ 4 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[52,2,7]] — qecdb.org 2BGA code, reconstructed from its H field

Direction & hypothesis

Track cell weight-4 × unrestricted, CSS board. The hypothesis was that the public parameter tables sit in front of this board: qecdb.org indexes 15,181 CSS codes with matrices, and sweeping all of them against the board's own 1,513 CSS entries should surface records whose parameters dominate an entry -- if the matrices survive reconstruction and the distance readings hold up. This candidate is record 67bc8f4b65e5efe366a705cb.

What was searched

  • Mirrored the whole CSS half of qecdb.org (/api/codes?cssOnly=1, 200 per

    • page): 15,181 records, n ≤ 300, d ≤ 17. A parameter pre-filter against the board left 1,527 records (dropped 136 over the verifier's weight-32 cap and 13,654 dominated on (n, k, d, w) as listed); every one of those was then fetched in full for its stabilizer generators.

  • Reconstruction: the record's H field is binary symplectic rows; split it

    • by rowspace (rank(A|B) = rank(A) + rank(B)) and keep the *lightest* independent basis of each side. Two traps this avoids -- a plain RREF inflates max check weight, and self-dual records store mixed (Y-bearing) generators of an otherwise CSS code. n, k and w were recomputed here; all 1,527 agreed with the database on k and on check weight.

  • Screening: every survivor at 50,000 RIS trials/side, bit-packed backend,

    • seed derived from the code's own fingerprint so a reading is reproducible. 608 records still beat a board entry; 314 of the 1,510 reconstructed survivors turn out to be direct sums, so connectivity is checked *before* anything is picked (104 of 221 remain).

  • Selection: cell-restricted advance (no dominator, at least one in-cell entry

    • beaten), ranked tracks (w ≤ 8), one code, a gain on something other than d, and no entry beaten on d alone (the d-only win is the inflation pattern). One record per (n, k, w) tuple.

  • This candidate: [[52,2,7]] at w = 4, record 67bc8f4b65e5efe366a705cb; qecdb description

    • "2BGA", database distance claim d = 7 (lower 7, upper 7).

Evidence trail

  • Ladder: 7 at deep2 → 7 at screen1

    • → packaging witness at 8000 trials/side (numpy lightest_logical, seed 0): X witness [7, 11, 13, 31, 33, 41, 51] and Z witness [8, 12, 31, 32, 40, 45, 49], both of weight 7.

  • Claim, precisely: a witness-backed upper bound d ≤ 7. Nothing here

    • certifies the exact distance; verify/validate_candidate.py also ran its own refutation search on the packaged document (seed recorded in the verdict beside this candidate) and found no lighter logical.

  • Board comparison (cell-restricted): beats 2 entries on n: 58-2-7.json, 63-2-7.json. Efficiency k d² / n = 1.88.
  • Verdict: passed: true, not refuted, not a duplicate of a board entry;

    • labels: advances the weight-4 x unrestricted board; literature novelty UNVERIFIED.

  • The full validator verdict (refutation seed, fingerprint, cell, labels) is

    • kept alongside the submission in this run's staging area for the reviewer.

Dead ends

  • Direct sums: 314 of 1,510 reconstructed records are not one code (the

    • verifier's tanner_connected / stabilizer_group_connected reject them). Ten of the first 35 picks fell here; their 947 blocks were re-screened as separate candidates.

  • d-only wins: candidates whose only strict axis over some entry was d were

    • rejected as distance inflation (three already-packaged ones were unstaged for this: [[48,18,4]], [[56,6,8]], [[84,44,3]]).

  • A second [[144,12,12]] w = 6 record passed the gate but ties the board's own

    • entry on all four axes, so it was unstaged: same parameters, no frontier gain.

  • ECZ (errorcorrectionzoo.org/all, 1,185 entries) is an index of parameters

    • and references only -- no matrices, so it cannot feed this pipeline; it is still the right place to check literature novelty later.

Tools

opencode/mimo-v2.6-flash-free (opencode) driving the repo's own kit (research/kit/: css, surrogate, submit, search) and the untouched verify/ stack; the bit-packed C++ RIS backend behind distance_rand(backend="fast") for the screens, numpy for the packaging witnesses. Whole scrape: ~25 minutes wall clock on a 10-core laptop.

Reproduction

1. GET https://qecdb.org/api/codes?cssOnly=1&pageSize=200&page=N for the list, GET https://qecdb.org/codes/67bc8f4b65e5efe366a705cb (the API path /api/codes/67bc8f4b65e5efe366a705cb) for the generators. 2. Parse H into binary symplectic rows; set A = rows with no Y, B = rows with Y only in their X half ... in practice: build the [A|B] matrix and split by rowspace, taking the lightest independent basis of each part (rank over GF(2), greedy lightest-first). 3. Verify H_X H_Zᵀ = 0, recompute k = n − rank(H_X) − rank(H_Z), take w = max row weight. 4. Screen: `distance_rand(HX, HZ, trials=50000, seed=int(fingerprint,16) % 2**31, backend="fast")`, then the same at 2,000,000 trials, then package with submit.make_submission(HX, HZ, trials=8000, seed=0). 5. Run verify/validate_candidate.py on the packaged document; keep only passed: true.

Parity checks

X-checks 25 (max weight 4) · Z-checks 25 (max weight 4)
H_X (25 checks, sparse supports)
[10, 25, 38, 50] [13, 23, 33, 37] [0, 18, 38, 51] [7, 20, 26, 28] [1, 21, 29, 46] [19, 22, 30, 47] [8, 23, 31, 48] [11, 22, 36, 44] [15, 25, 40, 45] [2, 16, 34, 41] [3, 17, 35, 42] [4, 18, 36, 43] [9, 24, 32, 49] [5, 7, 37, 44] [1, 6, 33, 39] [12, 19, 27, 40] [0, 8, 41, 45] [9, 20, 34, 42] [10, 21, 35, 43] [14, 24, 27, 39] [2, 11, 26, 51] [3, 13, 28, 46] [4, 14, 29, 47] [5, 15, 30, 48] [6, 16, 31, 49]
H_Z (25 checks, sparse supports)
[3, 20, 28, 42] [15, 19, 30, 40] [3, 21, 35, 46] [4, 22, 36, 47] [5, 23, 37, 48] [6, 24, 39, 49] [12, 25, 40, 50] [1, 14, 29, 39] [9, 17, 32, 42] [10, 18, 38, 43] [7, 11, 26, 44] [1, 13, 33, 46] [0, 2, 41, 51] [14, 19, 27, 47] [8, 15, 45, 48] [9, 16, 34, 49] [10, 17, 35, 50] [11, 18, 36, 51] [7, 13, 28, 37] [8, 16, 31, 41] [4, 21, 29, 43] [5, 22, 30, 44] [6, 23, 31, 33] [12, 24, 27, 32] [0, 25, 38, 45]
Code ID 52-2-7 · download JSON · raw on GitHub